Review of International Economics, 8(4), 597–613, 2000
Generalizing the Stolper–Samuelson Theorem:
A Tale of Two Matrices
Peter Lloyd*
Abstract
Past attempts to generalize the Stolper–Samuelson theorem have used a matrix of real income terms which
are sufficient but not necessary to define a change in utility. One can define a second matrix of terms which
are necessary and sufficient for a change in indirect utility. Using this matrix, the paper extends the
Stolper–Samuelson theorem to a model of any dimensions and to households which have diversified
ownership of factors. The theorem states that there is a positive and a negative element in every row and
every column of the matrix showing household responses to changes in goods prices.
1. Introduction
Even after its Golden Jubilee in 1991, there is still doubt about the generality of the
Stolper–Samuelson theorem (Deardorff and Stern, 1994). This is unfortunate. The
theorem is one of the few comparative statics propositions in general equilibrium
theory and it is the foundation of political economy models of tariffs and other taxes
and government interventions. The importance of the theorem derives from its key
message; goods price changes necessarily create conflict between households owning
different factors.
This paper considers a general model of an economy with constant returns to scale.
The introduction of diversification in households’ ownership of factors changes the
relationships between prices and real incomes fundamentally. Nevertheless, a generalization of the Stolper–Samuelson theorem that holds much more widely than earlier
versions can be obtained.
The paper begins with a brief history of attempts to generalize the original
Stolper–Samuelson theorem. The outcome is rather dismal. These extensions apply
to models in which the households are completely specialized in their ownership of
factors, as in the original theorem, and they require quite severe restrictions on the
technology when they do hold. All of these generalizations are special cases of one
generalized Stolper–Samuelson matrix. Then the paper presents the generalization of
the Stolper–Samuelson theorem. This uses a second matrix defined in terms of real
income effects which are both necessary and sufficient for a change in welfare. This
criterion of real income was introduced by Cassing (1981). It can be extended to
models of any dimensions beyond 2 ¥ 2 and to diversified household ownership of
factors. These results are interpreted in terms of a household’s excess demand vector,
using the notion of the imputed output vector of households. Some examples are given.
The paper ends with some concluding remarks.
* Lloyd: University of Melbourne, Parkville, Vic. 3052, Australia. Tel: 61 3 93448015; Fax: 61 3 9349 2397;
E-mail: [email protected]. I would like to acknowledge the comments of Alan Deardorff,
Rod Falvey, Murray Kemp, and Viktor Zolotenko on an earlier draft of the paper, and the comments of a
referee.
© Blackwell Publishers Ltd 2000, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA
598
Peter Lloyd
2. Past Generalizations
Stolper and Samuelson (1941) stated a theorem which predicted the movement of real
incomes of factors in an internationally trading economy that may protect an industry
and, in effect, certain factor owners employed in the industry. At the heart of this
version is a simple and strong relationship between goods prices and factor prices “that
has nothing to do with factor scarcity or abundance and is independent of whether
prices change because of protection or for any other reason” (Deardorff, 1994, p. 12)).
Deardorff called this the “essential” version and it is the version now used by most
economists. It does, however, require an open economy so that the disturbance which
causes the price change can originate in the rest of the world and one can maintain
the assumption that the technology and the preferences of the country do not change
with the change in price.
The theorem was proven for an economy in which there were only two goods and
two nonspecific factors. The factors are called labor and capital and the goods 1 and 2.
The factors are owned by separate groups of households. With constant returns to scale
and both goods produced, there are zero profits. We are interested in the change in
real incomes of the owners of the two factors when the prices of goods 1 and 2 change.
As the criterion of real income, Stolper and Samuelson chose the income of a factor
owner deflated by the price of good 1 or good 2. Since the factor endowments are fixed,
income is proportional to the price of the owned factor.
Assuming all goods are produced, there is a one-to-one mapping from goods prices
to factor prices which is given by the zero-profit conditions alone. Differentiating these
conditions in the manner introduced by Jones (1965), we get the equations
ÈqL1
ÍÎq
L2
qK 1 ˘ Èwˆ ˘ È pˆ 1 ˘
=
.
qK 2 ˙˚ ÍÎ rˆ ˙˚ ÍÎ pˆ 2 ˙˚
(1)
This follows the standard notation. (w, r) and ( p1, p2) are the vectors of prices of the
two factors and the two goods, respectively. The circumflex symbol denotes a proportionate change in a variable, and q is the transpose of the matrix of factor shares. In
matrix form, qŵ = p̂, where ŵ and p̂ are the column vectors of changes in the prices of
factors and goods. Solving for the changes in factor prices, we have ŵ = q-1p̂. If only
one goods price changes at a time, we get the matrix showing the changes in real
incomes in terms of prices of the good whose price has changed:
Èwˆ pˆ 1 wˆ pˆ 2 ˘
S=Í
= q -1 .
Î rˆ pˆ 1 rˆ pˆ 2 ˙˚
(2)
This may be called the Stolper–Samuelson matrix. If an element is greater than +1,
the real income of the factor must have increased; and if it is less than 0, the real
income of the factor must have decreased. This gives a criterion which is sufficient to
determine the direction of change in real incomes independently of factor owners’
preferences over goods.
It is easily shown that, after a suitable numbering of the columns and rows, the sign
matrix must have the form
È+ - ˘
sign S = Í
.
Î- + ˙˚
© Blackwell Publishers Ltd 2000
(3)
GENERALIZING THE STOLPER–SAMUELSON THEOREM
599
If an element is positive, it is greater than unity. Hence, there is only one permissible
sign pattern, that in which the diagonal elements are positive (and greater than unity)
and the off-diagonal elements are negative. Moreover, the positive elements correspond to changes in the real incomes of the factors which are used intensively in the
production of the good whose price has changed. This proves that, when the price of
one good increases, the real income of the factor used intensively in the production of
this good must increase and that of the other factor must decrease. There is, therefore,
conflict between the factor owners. One group of factor owners would support the
increase in the price of one good and the other would oppose it.
Over the last 50 years numerous extensions of the Stolper–Samuelson theorem have
been made. They are all special cases of the following matrix:
È wˆ 1 pˆ 1
S=Í
Îwˆ m pˆ 1
. . . wˆ 1 pˆ n ˘
.
. . . wˆ m pˆ n ˙˚
(4)
This might be called the generalized Stolper–Samuelson matrix as the elements are
the same income terms as used in the 2 ¥ 2 version of the theorem. This m ¥ n matrix
holds for m factors and n goods. It allows for various cases in terms of the number of
factors and goods.
When the number of goods and factors is uneven, univalence breaks down. Attempts
to generalize the theorem to higher dimensions, therefore, have been mainly confined
to dimensions of n ¥ n. It is usually assumed that there are no intermediate inputs.
This can be relaxed to include intermediate usage of the produced outputs, but pure
intermediates can be admitted only if other produced commodities are omitted as this
would otherwise upset the evenness of the model. But even this n ¥ n case has proven
extraordinarily difficult.
In this case, let w = (w1, . . . , wn)T and p = ( p1, . . . , pn)T be the column vectors of
prices of factors and goods, respectively. Assuming again that all goods are produced,
we have qŵ = p̂ and, hence, we have the n ¥ n Stolper–Samuelson matrix
È wˆ 1 pˆ 1
S=Í
Îwˆ n pˆ 1
. . . wˆ 1 pˆ n ˘
= q -1 .
. . . wˆ n pˆ n ˙˚
(5)
A problem now is that there is no natural definition of factor intensity when the
number of goods and factors is greater than 2. Hence, there is no natural way of characterizing the sign matrix to represent generalizations of the Stolper–Samuelson
theorem. Three principal generalizations have been proposed:1
1. The “strong” generalization: Each of the diagonal elements is positive (and greater
than unity) and each of the off-diagonal elements is negative.
2. The “weak” generalization: Each of the diagonal elements is positive (and greater
than unity).
3. The “basic” generalization: In every column and every row there is at least
one element which is positive (and greater than unity) and at least one which is
negative.
The first two generalizations were proposed by Chipman (1969). The third was first
considered by Meade (1968), Ethier (1974), and Jones and Scheinkman (1977).
Both the strong and the weak generalizations involve an association between goods
and factors which is mutually exclusive and exhaustive. For every good, there is one
and only one factor associated with this good such that the increase in the price of the
© Blackwell Publishers Ltd 2000
600
Peter Lloyd
good must increase the real income of the associated factor. Hence, these generalizations preserve the identification of the particular factor which is associated with each
good, as Stolper and Samuelson were able to do for the 2 ¥ 2 case in terms of the factor
intensities of the goods.
By contrast, the third generalization does not associate factors with goods. If it holds,
it might not be possible to array the elements so that every column has a positive
element on the diagonal. It is, therefore, weaker than the “weak” generalization in this
sense, even though it does specify the presence of at least one negative element in each
column whereas the “weak” generalization does not. The column property means that
when the price of any one good changes, at least one factor has an increase in real
income as defined by Stolper and Samuelson, and at least one has a decrease in real
income. There must be conflict between groups of factor owners. The row property
states that, for every group of factor owners who derive their income solely from the
factor in this row, there is at least one good whose price increase will raise their real
incomes and at least one other good whose price increase will lower their real incomes.
This generalization is thus the minimal restriction that preserves the conflict features
of the original theorem. It has the added advantage that it extends to nonsquare
matrices when the number of goods and factors is uneven.
None of the three generalizations holds for the 3 ¥ 3 version of the model without
restrictions on the technology. Kemp and Wegge (1969) provided a factor intensity condition which is sufficient for the strong generalization, and Chipman (1969) provided
a factor intensity condition in terms of a dominant diagonal which is sufficient for the
weak generalization. No condition which is sufficient (or necessary and sufficient) for
the third generalization has been provided to my knowledge.
A number of authors have examined the case when n > 3 but the number of factors
and goods is still even. Uekawa (1971) obtained sufficient conditions for the weak
generalization, and Uekawa (1971) and Shimomura (1997) obtained necessary and
sufficient conditions for the strong generalization. Shimomura showed that the
Kemp–Wegge condition is necessary for the strong generalization to hold. However,
these results involve quite stringent generalized factor intensity conditions. Some
further results have been obtained by Jones and other coauthors (Jones and Mitra,
1995, and references therein). These involved other restrictions on the factor intensities between sectors using the benchmark share rib of factor shares. The third or basic
generalization does not hold in the case of n > 3. Ethier (1974) was able to prove the
desired column property. Using this result, Jones and Scheinkman (1977) were able to
prove that every row must have a negative element but could not prove that it had an
element greater than unity. Moreever, Ethier’s proof requires the rather extreme
assumption that all qij > 0; that is, all of the factors are used in all industries.
For the m ¥ n case, Jones and Scheinkman showed that every column has a negative element under the weaker condition that every factor is used in strictly positive
quantities by at least two industries and every industry employs strictly positive
amounts of at least two factors. While this is not a very strict condition, the first part
rules out industry-specific factors.
The specific factor model developed formally by Jones (1971) for the model of 3 ¥
2 dimensions and extended by Jones (1975) to (n + 1) ¥ n is an example of m ¥ n and
has been widely used in the political economy literature. In this model, every column
does have an element greater than unity and an element less than zero. Moreover, if
one considers the square submatrix excluding the row for the nonspecific factor, this
has the pattern of the strong generalization. In effect, factor specificity is a form of
association between the factor and the good. As with the Kemp–Wegge condition in
© Blackwell Publishers Ltd 2000
GENERALIZING THE STOLPER–SAMUELSON THEOREM
601
the 3 ¥ 3 model, it generates the pattern of the strong generalization for these inputs
and it identifies the input which gains when each price is increased.
But the row relating to the nonspecific factor has neither an element greater than
unity nor an element less than zero. Ruffin and Jones (1977) referred to the absence
of an element greater than unity in this row as the “neoclassical ambiguity.”Thus, again,
the main sticking point in the basic generalization is the property that the row must
have an element greater than unity.
Using a different real income criterion from Stolper–Samuelson and earlier writers,
Cassing (1981) re-examined the case of n ¥ n dimensions. He established the row property of the basic generalization which Jones and Scheinkman could not establish. He
did so without restricting the technology. This is an important but generally neglected
paper. His criterion of real income change is employed below.
All of these results have assumed that all households own only one factor. They have
also assumed that the number of goods produced is equal to the number consumed,
which rules out pure export and import goods. Plainly, we want a proposition relating
real income changes to price changes which holds much more generally. I now
consider the possibility of extending the theorem for a much more general model.
3. Generalization of the Stolper–Samuelson Theorem
Consider a small open economy. There are any number of goods (n) and factors (m)
and households (H ). A good must be consumed or produced but there may be goods
consumed which are not produced and vice versa. Factors are fixed in supply but may
be specific or nonspecific. There may be intermediate inputs used in the production of
goods. Households own the national endowments of factors and a household may own
any number of the factors. Let the goods be indexed by i = 1, . . . , n, the factors by
j = i, . . . , m and the households by h = 1, . . . , H.The vector of prices of goods is denoted
by p, the vector of prices of factors by w, and the vector of endowments for the
economy by v.
On the supply side, there is a national product function, G( p, v), for a competitive
economy with a technology that is nonjoint and has constant returns to scale in all
industries.2 It is well-known that Gp = y( p, v) and Gv = w( p, v), where y is the vector
of industry outputs. On the demand side, every household h has an indirect utility function, Vh( p, Ih), where Ih is the income of the household. In general equilibrium, the
income of the household is endogenous. Utilizing w = w( p, v), the household income
function is Ih = w( p, v)vht = gh( p, v, vh).
Let prices change only one at a time. Differentiating the indirect utility function with
respect to the price of good i, one has the expression for welfare change
Vˆ h pˆ i ∫ ( pi [∂ V h ∂ I h ◊I h ])(∂V h ∂ pi ).
(6)
These expressions can be arrayed in an H ¥ n matrix
È Vˆ 1 pˆ
Vˆ 1 pˆ n ˘
V =Í H 1
˙.
ÎVˆ pˆ 1 Vˆ H pˆ n ˚
(7)
The sign of the element (V̂ h/p̂i) provides the criterion of change in real income of the
household. If an element of this matrix is positive, the household concerned has a
higher real income when the price of the good in the column increases. If the element
is negative, the household concerned has a lower real income when the price of the
good in the column increases. This criterion requires knowledge of the pattern of
© Blackwell Publishers Ltd 2000
602
Peter Lloyd
demand of each household. But there is no ambiguity; an element greater (less) than
0 is necessary and sufficient for an increase (decrease) in the real income (= utility) of
the household.
The matrix in equation (7) differs from the generalized Stolper–Samuelson matrix
in equation (4) in terms of the criterion of a change in real income and in the diversification of the income of households. It permits any pattern of diversification.
The three generalizations of the Stolper–Samuelson theorem can be restated in
terms of this V matrix. The strong generalization holds, for a square matrix, if and only
if all elements on the diagonal are strictly positive and all off-diagonal elements are
strictly negative. The weak generalization holds, for a square matrix, if and only if all
elements on the diagonal are strictly positive. The basic generalization holds for a
matrix of any dimensions if and only if there is a strictly positive and a strictly negative element in each row and column.
Now, differentiating Vh( p, Ih) and using Roy’s identity, we have
Vˆ h pˆ i = Iˆ h pˆ i - f ih,
(8)
h
i
where f is the share of the good i in the budget of household h. In equation (8), the
welfare effect of changes in the price of a single good contains two terms, one of which
measures the effect of the price change on the household as an income-earner and one
of which measures the effect on the household as a consumer. The criterion of real
income used by Stolper and Samuelson and the subsequent literature neglects the
second term.
Ih is endogenous in the general equilibrium. Household income is the income from
the factors which the household owns, Ih = wvht, where vht is the column vector of factors
owned by the household, the transpose of vh. Utilizing the mapping w = w( p, v), the
household income function is
I h = w( p, v)v ht = g h ( p, v, v h ).
(9)
These functions are homogeneous of degree +1 in p. Differentiating, we have
(Iˆ h
pˆ i ) = Â a hj (wˆ j pˆ i ),
j
Âa
h
j
= 1,
(10)
j
where the weights a hj = wjv hj/Sjwjv hj are shares of income earned from each factor.
Substituting this expression in equation (8), we now have
Vˆ h pˆ i = Â a hj (wˆ j pˆ i ) - f ih ,
h = 1, . . . , H .
(11)
j
The introduction of diversified household ownership changes the properties of the
model fundamentally. Consider the case in which vh = bv; that is, each of the H households owns an equal fraction ( b = 1/H ) of the nations’ endowment vector, and all
households have the same preferences. Hence, all households are equally (and completely) diversified. There is then no diversity among households in either the pattern
of factor ownership or the pattern of consumption. In (8), the term Î h/p̂i = Ŷ/p̂i, where
Ŷ is the change in national income. In each column of V the income terms for all households are identical. Similarly, every household has the same consumption term fhi = fi
Therefore, V is a matrix in which all terms in a column have the same sign. This holds
for all q.
Diversity is a departure from equal ownership of factors and equal consumption
shares among households. With any diversity, the terms in a column will cease to be
© Blackwell Publishers Ltd 2000
GENERALIZING THE STOLPER–SAMUELSON THEOREM
603
equal. With small diversity in household ownership of factors and/or preferences, the
terms in the columns of V will be the same sign for all i.
Evidently, the assumption that each household owns only one factor that is not
owned by any other household, which is built into the previous literature, is extreme
diversity in factor ownership and produces atypical results. All of the results reported
in section 1 hold because of the combination of the assumed restrictions on technology and the restriction of household ownership to the extreme case of completely undiversified ownership.
We need to take diversification into account as the reality is that most households
today are at least partly diversified in endowments. Diversification is a property of
the endowment vector of individual households. It is a departure from completely
specialized single-factor endowment.
Theorem 1. Consider an m ¥ n ¥ H economy in which at least two goods are produced
and two factors are used. In the H ¥ n matrix of terms showing the change in real
income (= utility) of households in response to changes in the prices of goods one at
a time,
È Vˆ 1 pˆ
Vˆ 1 pˆ n ˘
V =Í H 1
˙,
ÎVˆ pˆ 1 Vˆ H pˆ n ˚
• there is at least one strictly positive element and at least one strictly negative element
in every row; and
• there is at least one strictly positive element and at least one strictly negative element
in every column for a good which is produced, if and only if there is sufficient diversity among households in their ownership of factors and/or their preferences.
Proof of the row property. The household real income is given by Vh( p, gh( p, v, vh))
= Wh( p, v, vh). Wh is homogeneous of degree 0 in p since V h is homogeneous of degree
0 in p and gh, and gh is homogeneous of degree +1 in p. Applying Euler’s theorem and
expressing the results in proportionate changes, we have
n
 Wˆ
h
pˆ i = 0.
(12)
i =1
The terms (Ŵ h/p̂i) are the elements in the hth row of the matrix of V. From (12), there is
at least one strictly positive element and at least one strictly negative element in every
row.3 (This ignores the very special borderline case in which the elements are all zero.)
Proof of the column property: (i) necessity. If there is zero diversity in both factor
ownership and preferences, every element of column i, V̂/p̂i = Î h/p̂i - fhi, must have the
same sign. Some diversity among households in factor ownership and/or preferences
is necessary for the change in a goods price to affect households differentially.
Proof of the column property: (ii) sufficiency. For any technology and any set of
household preferences, there exists a distribution or distributions of the national
endowments among households which yields the column property of the theorem.
Consider a column of V. Take two households, one of which owns a factor with
(ŵ j/p̂i) > 0 and one of which owns a factor with (ŵ j/p̂i) < 0. Call these households s and
t. For any set of fhi, we can increase the weight attached to the positive term
© Blackwell Publishers Ltd 2000
604
Peter Lloyd
(ŵ j/p̂i) for household s and decrease that attached to a negative term (ŵ j/p̂i) for household t until (V̂ s/p̂i) > 0 and (V̂ t/p̂i) < 0.
Sufficient diversity may come instead from diversity of preferences. Consider the
case in which the H households are equally diversified in terms of endowments.
Then V = (Ŷ/p̂i) - f, where (Ŷ/p̂i) is the matrix in which every element in column i is
identical as all share equally in the change in national income. (Ŷ/p̂i) > 0 when the
price of i is increased if the good is produced. Moreover, (Ŷ/p̂i) = (Ĝ/p̂i) = piyi /Sipiyi
(using the derivative property of G, Gp = y( p, v)).This lies in the unit interval. It requires
only a limited variation in the preferences of some households in terms of fhi, which also
lie in the unit interval, to ensure that there is a positive and a negative element in every
column.
It follows immediately from equation (8) that a change in the price of any good must
make at least one household better off and at least one worse off.
The proof of the column property assumes that there is some factor for which
(ŵ j/p̂i) > 0 and another factor for which (ŵ j/p̂i) < 0 when the price of good i increases.
Jones and Scheinkman (1977) showed that this holds for a Heckscher–Ohlin-type
model of any dimensions, provided every industry employs at least two factors
and every factor is employed by at least two industries. This result also holds for the
specific factor model in which specific factors, by definition, are employed in only one
industry. It holds generally if all industries employ at least two factors and, when the
price of good i rises, there is one factor at least in industry i whose price rises and which
is employed in another industry. This can be assumed to hold for all technologies.
One case which is ruled out is that in which the economy is completely specialized
in the production of only one good, say good 1. This good must be the export good. In
this case, (ŵ j/p̂1) = 1 for all j and every household must gain from increase in its price
and lose from an increase in the price of the import good (assuming that each household consumes at least two goods).4 More generally, an increase in the price of any
good which is imported and not produced at home will make all households worse off.
Another case which is ruled out is that in which there is only one factor. In this case,
the income of all households again changes by the same proportion as the change in
the good price and diversity in preferences is insufficient to produce both gainers and
losers.
The power of this result stems from the use of an income criterion which includes
the second component of a welfare change due to the consumption effect. Uekawa
(1971) provides an example of a technology which is instructive in the present context.
The matrix of factor shares and its inverse are
0
2 10 ˘
4 5
- 7 10 ˘
È8 10
È 9 10
Í
˙
Í
-1
q = 7 20 7 20 6 20 , q = - 21 10 24 5 - 17 10˙.
Í
˙
Í
˙
ÍÎ 0
ÍÎ 7 5
6 15 9 15 ˙˚
- 16 5
14 5 ˙˚
It is easily verified that the share matrix violates the Kemp–Wegge and the Chipman
sufficiency conditions. It also fails to satisfy the requirements of the third “basic” generalization. Therefore, this case is a counter-example to all three generalizations of the
Stolper–Samuelson theorem. However, in this case, the alternative matrix V is
4 5
- 7 10 ˘ Èf11
È 9 10
V = q -1 - f = Í- 21 10 24 5 - 17 10˙ - ÍÍf12
Í
˙
ÍÎ 7 5
- 16 5
14 5 ˙˚ ÍÎf13
© Blackwell Publishers Ltd 2000
f 21
f 22
f 23
f 31 ˘
f 32 ˙˙.
f 33 ˚˙
GENERALIZING THE STOLPER–SAMUELSON THEOREM
605
By inspection, this matrix has the sign pattern required for the basic generalization for
any 3 ¥ 3 nonnegative matrix of consumption shares. Thus, the basic generalization
holds in terms of V. The upholding of the basic generalization in terms of V but not S
is an expression of the sufficiency but nonnecessity of the Stolper–Samuelson criterion
of real income change.
Of course there are more than three households in an economy. With H π n, the
pattern of the “weak” and “strong” generalizations are now not defined. With many
households and diversified factor ownership, there will generally be many positive and
many negative elements in each column. Similarly, with multiple factors, there may be
more than one positive and more than one negative element in a row. The likely
patterns of the matrix V will be very different from the patterns of the matrix S.
There is another generalization concerning the effects on household incomes of
changes in factor endowments. There is a small literature on these qualitative properties of trade models, mainly derived from the literature on immigration (see Ruffin,
1984, pp. 259–65). In this case we are interested in the properties of the matrix
È Vˆ 1 vˆ
Vˆ 1 vˆ m ˘
Z=Í H 1
˙,
ˆ
ÎV vˆ1 Vˆ H vˆ m ˚
which holds when the aggregate endowments change but goods prices are now
constant. The household real income is still given by the indirect utility function
Vh( p, gh( p, v, vh)) = Wh( p, v, vh). Consider changes in the vector of aggregate endowments, v.We can derive results similar to those for changes in the vector of goods prices.
Wh( p, v, vh) is homogeneous of degree zero in v in a constant-returns-to-scale economy.
Therefore there is a positive and a negative element in every row of Z. Similarly, in
the same fashion as in Theorem 1, we can establish that there is a positive and a
negative element in every column, subject again to a weak restriction on the number
of goods produced and the number of factors used in the production of goods.
However, there are two differences from the results that apply when goods prices
change and factor endowments are constant. First, the case of zero elements in all rows
arises when m = n and n goods are produced. This is the Rybczynski theorem. Second,
an increase in the aggregate endowment of some factor must in fact be held by some
individual. We have, therefore, to add in the factor quantity effect to those households
which hold the additional endowment (dvh). If the increase is distributed among the
households which are affected negatively by the change in aggregate endowments,
some or all of these may now gain from the endowment change.
4. An Interpretation of the Theorem
The best interpretation of the conflict among households in the columns of V is to
regard households which gain and those which lose as being on opposite sides of the
market. This requires the concept of the imputed output vector of a household.
The outputs imputed to a household, denoted by ỹ h, may be calculated in the same
manner as outputs are determined for the national economy. (The concept of the
imputed output of the household has been used by Lloyd and Schweinberger (1988,
1997).) Each household owns a given endowment of factors which will be fully
employed. The household is assumed to possess the same technologies of producing
each good as the nations’ firms. Its output vector is given by
y˜ h = yvv h = y˜ h ( p, v, v h ).
(13)
© Blackwell Publishers Ltd 2000
606
Peter Lloyd
yv is the n ¥ m matrix of derivatives with respect to v of the output vector y( p, v).5
With this convention, the national output vector is the sum of the household vector.
We may, therefore, regard ỹ h as that part of the national output vector which can be
imputed to household h. Moreover, the value of the household’s output is equal to its
income; that is, pỹ h = wvht = gh( p, v, vh) (see Lloyd and Schweinberger, 1997). As each
household’s expenditure is equal to its income, there is balance-of-payments equilibrium for the economy. Note that g ph = ỹ h.
The economic meaning of this imputation can be obtained from inspecting the
imputed output by household of good i. By derivation, the ith element of ỹ h is
m
y˜ ih = Â ∂ yi ∂ v j v hj .
(14)
j =1
That is, the imputed output of good i by household h is the weighted sum of the inputs
supplied by this household to the firms of the economy. The weights are the marginal
effects of the change in the aggregate supply of this input on the aggregate economy
output of each good. These can be termed the Rybczynski weights.6 They may be positive or negative. Hence, the output of a commodity imputed to a household may be
positive or negative. If the household’s endowments lie outside the cone of diversification, one or more of the outputs imputed to the household will be negative. A household may produce more than the national output of a good if other households in the
aggregate produce a negative output of the good. The household’s output vector is
merely the vector it would be required to produce if its endowments are to be fully
employed and it uses the national technology. It is a notional concept which will enable
us to obtain further results.
Finally, with this concept of the imputed outputs of households, we can now define
the excess demand of household h for good i as e hi = x hi - ỹhi , where x hi is the demand
by household h for good i. Now ỹhi = ỹ h( p, v, vh) and x hi = x hi ( p, gh( p, v, vh)). Therefore,
e hi = e hi ( p, v, vh). From the derivation of imputed household outputs, we have, for each
good, She hi = ei, the excess demand in the economy for good i.
This definition of the excess household demand can be related to the change in
the utility of a household when the price of a good changes. Using Roy’s identity and
the derivative property of the household income function (g ph = ỹ h) used in equation
(6) yields
dV h dpi = ( y˜ ih - xih ) ∂V h ∂ I h
> (<) 0 as ( y˜ ih - xih ) > (<) 0,
(15)
since ∂Vh/∂Ih > 0. A household gains or loses from an increase in the price of a good,
depending on whether it is an implicit net seller or buyer of the good.7 Alternatively,
this result can be expressed in terms of proportionate changes:
Vˆ h pˆ i = g ih - f ih ,
(16)
where g hi = piỹhi /Sipiỹ hi is the share of the income earned from the imputed production
of good i, fih = pix hi /Sipix hi is the share of the expenditure of household h on commodity i. The matrix V can be written succintly as
V = g - f,
(17)
where g is the H ¥ n matrix of terms (g hi ) and f is the H ¥ n matrix of terms (f hi ).
© Blackwell Publishers Ltd 2000
GENERALIZING THE STOLPER–SAMUELSON THEOREM
607
Example 1: The Original Stolper–Samuelson Model
The concept of imputed output can be used to verify the original Stolper–Samuelson
theorem for the 2 ¥ 2 model. The Rybczynski equations relating commodity outputs
to factor endowments are
aL1 (w, r ) y1 + aL 2 (w, r ) y2 = L,
aK 1 (w, r ) y1 + aK 2 (w, r ) y2 = K .
(18)
The solution to these equations is, for given commodity and factor prices:
y1 = [aK 2 L - aL 2 K ] D,
y2 = [aL1 K - aK 2 L] D,
(19)
where D = aL1aK2 - aL2aK1 is the determinant of the system of equations. Hence the
Rybczynski weights are
È ∂ y1 ∂ L ∂ y2 ∂ L ˘ È aK 2 D -aK 1 D˘
R=Í
=
.
Î∂ y1 ∂ K ∂ y2 ∂ K ˙˚ ÍÎ-aL 2 D aL1 D ˙˚
(20)
For household h, the ith element of the vector ỹhi is ỹhi = S 2j=1∂yi/∂vjv hj. In each
column and row of the matrix of Rybczynski weights, R, there is one positive and
one negative element. Thus, when the stock of one factor increases, the output of the
good which is intensive in this factor increases and the output of the other good
decreases.
Suppose now that there are two households (or groups of households), one of
which owns the labor supply of the economy and the other of which owns the
capital stock. Let the household which owns labor be household 1 and that which
owns capital household 2. Let the good which is intensive in the factor labour be
good 1 and the other be 2. Using equation (19) above, the implicit outputs of the
households are y11 = ∂y1/∂L.L, y12 = ∂y2/∂L.L, y12 = ∂y1/∂K.K, and y22 = ∂y2/∂K.K. Each
household produces a positive quantity of the good which uses intensively the factor
with which it is endowed and a negative quantity of the other. Since the sum of
the values of the outputs of the two goods by a household is equal to its income,
the household which has a positive imputed output of the good has an imputed income
from the production of this good which is greater than its household income.
Consequently, this household has a value of ( ỹ hi - x hi ) which must be positive no
matter what proportion of the budget of this household is spent on the good. The
corresponding term for the other household must be negative no matter what its
expenditure pattern. This proves the original Stolper–Samuelson theorem by a quite
different approach.
Example 2: The Specific Factor Model
This is an important example as the model is uneven; there are three factors and only
two goods. Jones also assumes that there are three completely undiversified factor
owners, each of whom owns all of the nations’ stock of one of the three factors. The
nonspecific factor is called labor, L, and the specific factors are the capital specific
to industry 1, K1, and the capital specific to industry 2, K2. The Rybczynski (fullemployment) equations relating the outputs of the two goods to the endowments of
the three factors are given by
© Blackwell Publishers Ltd 2000
608
Peter Lloyd
aL1 (w, r1 , r2 ) y1 + aL 2 (w, r1 , r2 ) y2 = L,
aK 1 (w, r1 , r2 ) y1 = K1 ,
aK 2 (w, r1 , r2 ) y2 = K 2 ,
(21)
where r1 and r2 are the prices of K1 and K2 respectively. The factor prices as well as the
outputs, y1 and y2, are unknown. Consequently, as is well known, determination of the
outputs requires the two equations which relate factor prices to output prices. This
gives five equations to determine the five unknowns ( y1, y2, w, r1, r2). The Rybczynski
weights can now be obtained by differentiating totally the five equations. This matrix
has the sign pattern8
È ∂ y1 ∂ L
Sign Í∂ y1 ∂ K1
Í
ÍÎ∂ y1 ∂ K 2
∂ y2 ∂ L ˘ È+ + ˘
∂ y2 ∂ K1 ˙ = Í+ -˙.
˙ Í
˙
∂ y2 ∂ K 2 ˙˚ ÍÎ- + ˙˚
(22)
Using this matrix in equation (14), one sees immediately that a household which
owns a specific factor produces a positive output of the good which uses this specific
factor and negative output of the other good, whereas the household owning the nonspecific factor produces both goods positively. Furthermore, a household owning one
of the specific factors actually produces more than the national output because the
Rybczynski weights are greater than the terms (1/aKi) = yi in equation (22). In this
model, an increase in the stock of a factor causes the factor price to decrease and this
in turn increases the input of this factor at the margin. Hence, the square submatrix in
the second and third rows of the V has precisely the same sign pattern as the original
Stolper–Samuelson 2 ¥ 2 model.
These results for the 2 ¥ 2 and the 3 ¥ 2 models and the use of imputed outputs generally can be explained by the “reciprocity relation.” For a competitive equilibrium
such that all goods are produced and all factors fully employed, as we have been assuming, the reciprocity relation states that yv = wp. That is, ∂yi/∂vj = ∂wj/∂pi for all i and j.
This explains why the imputed outputs are linked to the terms (wj/p̂i)which appear in
the S and V matrices.
With this concept of imputed household outputs, a new interpretation can be given
to gainers and losers from a price rise. One can regard households which gain and those
which lose as being on opposite sides of the market. It is possible for all households
to be on the same side of the market and consequently for conflict between households to be absent when the price of the good changes. However, a strong presumption of conflict between households remains in an economy with diverse individual
households. This must occur if there is sufficient diversity among households.
A 3 ¥ 3 example will demonstrate the role of imputed outputs in the determination
of the patterns of factor ownership in generalizations of the theorem.
Example 3: 3 ¥ 3 Heckscher–Ohlin Model
It is convenient to use the Leamer triangle. This triangle has been used previously in
the analysis of the Stolper–Samuelson theorem by Jones (1992) and Lloyd and
Schweinberger (1997). In this application, the triangle is applied to the endowments
of the individual households within the national economy rather than the endowments
of the nations within the world economy as in Leamer (1987). It is obtained by the
intersection of the positive orthant of the 3-dimensional factor space with a unit-value
© Blackwell Publishers Ltd 2000
GENERALIZING THE STOLPER–SAMUELSON THEOREM
609
plane. The triangle is then the unit simplex. The triangle represents graphically in
2-dimensional space the relative proportions of the three factors.
Thus, the endowment vectors of the individual households are points in the triangle. For a household, the coordinates of these points are the shares of the income of
the household from each factor (a 1h, a h2 , a 3h). The endowment points of the households
are distributed over the triangle. A household which is completely diversified in that
it owns a strictly positive quantity of each factor will have an endowment point in the
interior of the triangle. A household which owns two of the three factors will have an
endowment point on a side of the triangle, and one which is completely undiversified
will have it at a vertex.
We also plot in the triangle the vector of cost-minimizing input coefficients used
in the production of each good, qi. The coordinates of these points represent the distributive shares of the factors, (q1i, q2i, q3i). These vectors represent the technology.
For any technology, the Leamer triangle can be partitioned into seven zones showing
the pattern of complete, partial, or zero diversification in production by a household
whose endowment point is in this zone. The zone of complete diversification is the area
within the cone of diversification. If a household’s endowment vector lies in this zone,
the household will produce all three goods in strictly positive quantities. If it lies in
another zone, it will produce positive quantities of one or two goods only. This partition holds for any technology. If q lies on a side of the triangle (because one of the
factors is not used in the production of one good, good i), the zone in which only good
i is produced positively reduces to the point qi.
Figure 1 provides an illustration. The technology in this figure is an example of
a technology which satisfies the Chipman dominant-diagonal condition. The points
labelled q1, q2, and q3 show these sets of coefficients for goods 1, 2, and 3, respectively.
The lines connecting these points form the familiar cone of diversification. For the
technology given by the points q1, q2, and q3, the zones of zero, partial, or complete
Figure 1. The Heckscher–Ohlin Model
© Blackwell Publishers Ltd 2000
610 Peter Lloyd
diversification in the imputed outputs of households are shown. The goods produced
in each zone are given in parentheses.
First assume, as in the original Chipman case, that there are three households which
are completely undiversified in their factor ownership. Let households 1, 2, and 3 own
the stocks of factors 1, 2, and 3, respectively. The endowment points of the three households are at the three vertices of the triangle. Then, households 1 and 3 produce only
one good in positive quantities, goods 1 and 3, respectively. These are the goods which
use intensively the factor which the household owns. The other two goods are produced in negative quantities. Hence, the value of the imputed output of the respective
good is greater than the income of these two households. These two households gain
from an increase in the price of this good and lose from an increase in the prices of
the other goods. Household 2 produces good 3 as well as good 2 in positive quantities
and good 1 in a negative quantity. As q2 dominates the other terms in the second
column of q, the value of the output of this good is larger than the income of
household 2. The weak generalization of the Stolper–Samuelson theorem follows from
equation (16).
Allowing diversification of household ownership of factors opens up a much richer
variety of possibilities. For all sets of endowments in the zones marked (1), (2), and
(3), the strong generalization of the theorem holds. Since only one good is produced
positively in each household, the value of the imputed output of this good exceeds its
income and the household gains from an increase in the price of the good, irrespective of its preferences. For the other two goods, g hi < 0 and, therefore, (g hi - fih) < 0.
(These endowments are such that the strong generalization holds even though the
Kemp–Wegge condition does not hold.)
However, with diversity in factor ownership, it is unlikely that the patterns of
the weak and strong generalizations will hold, even for an n ¥ n model and even if the
technology yields the required pattern of Stolper–Samuelson “real income” effects for
this model. Diversified households will be affected by more than one factor price
change.
One can find a region of household diversification which yields the basic generalization in section 4. The patterns of diversification in this region will suffice for the generalization, irrespective of the household preferences. The basic generalization holds
for all sets of endowments such that each household produces one good in value
greater than its total output (income). This holds for any technology as represented by
a q. These regions of factor ownership are sufficient for the theorem. With knowledge
of the preferences of the households, there is a wider set of household distributions
which is necessary and sufficient for the theorem in section 4. Even in the extreme case
in which each household owns a fraction of the nation’s endowments, all households
will have the same endowment point in the cone of diversification but the theorem will
hold if there is sufficient diversity in preferences.
Example 4: Complete Specialization
It was noted in section 4 that the column property of the theorem does not apply in
the case of complete specialization. In this case the imputed output of the import good
is zero for all households. Consequently a household is adversely affected by an
increase in the price of the import good or, if it does not consume the good, it is unaffected. That is, all households are on the same side of the market. Note, however, that
this result does not apply in reverse to a good which is produced but not consumed.
In this case, an increase in its price will change production and factor prices and there© Blackwell Publishers Ltd 2000
GENERALIZING THE STOLPER–SAMUELSON THEOREM
611
fore household incomes. Under the assumptions made, some households will gain and
some will lose.
5. Concluding Remarks
The new theorem retrieves the spirit of the Stolper–Samuelson theorem. It has not
been stated previously to my knowledge. Cassing (1981) proved the row property
for the restricted model with specialized factor ownership and n ¥ n dimensions. If
the numbers of households and goods and factors is n and every household owns
only one factor and every factor is owned by one household, each household can be
associated with a factor. Thus, households can be numbered in the same way as factors.
Then, for any household h, V̂ h/p̂i = ŵh/p̂i - f hi . The matrix of real income changes is
V = q-1 - f. Since q-1 and f are both row stochastic, it follows immediately that the row
sums of V are zero. Hence, there must be at least one positive and one negative element
in each row of the matrix. Lloyd and Schweinberger (1988, p. 281) stated the general
form of the row property in the theorem, using the household trade expenditure
function.
The general proof of the row property derives from the use of a true index of utility
rather than the Stolper–Samuelson criterion of real income change. In fact, it is the
homogeneity property of the indirect utility function which produces the result that
Cassing noted. This result is much more general than the special case of an n ¥ n
economy with undiversified households which Cassing considered. It applies to any
household with a diversified factor ownership and it allows specific as well as nonspecific factors, intermediate inputs and nonproduced goods. The column property has not
been proven before.
Thus, it turns out that it is the row property which holds generally, not the column
property. This is the opposite of the conclusion of Ethier (1974) and Jones and
Scheinkman (1977) for the case of households which are completely undiversified in
terms of the factor ownership and using the Stolper–Samuelson criterion of real
income change. However, this column property still requires a Stolper–Samuelson-type
conflict between at least two factors used in the production of at least two goods. The
2 ¥ 2 model is the minimum dimension for universal conflict between households.
The contrast between these results and those of earlier generalizations of the
Stolper–Samuelson theorem is the contrast between the two matrices in equations (4)
and (7). Equation (7) permits any pattern of diversification in factor ownership, uses
the necessary and sufficient criterion for a utility change, and is unrestricted in terms
of the dimensions of the model.
The only restrictions which rule out a part of the gains from international trade is
that the number of goods which are consumed is fixed (but not necessarily equal to
the number of goods produced) and there are no economies of scale. This is unlike the
models of Krugman, Helpman and others in which this number is endogenous and
there are economies of scale. Even here, one can note that the consumers in the
Krugman–Helpman models are identical in terms of their preferences (see, for
example, Helpman and Krugman (1985), especially chapter 9.5). Again it is this lack
of diversity which accounts for the result that all households may gain from trade, not
the presence of economies of scale and product varieties.
In deriving the Stolper–Samuelson-type theorem above, the general household
income function is an expeditious method of extending the column property of the
matrix of household changes in real income. However, the crucial step in the extension of both the row and column properties of the Stolper–Samuelson theorem is the
© Blackwell Publishers Ltd 2000
612
Peter Lloyd
use of the true index of utility change. This provides a criterion which is necessary and
sufficient for a price index to increase or decrease the real income of a household.
The Stolper–Samuelson criterion was weaker and sufficient for the simple 2 ¥ 2
Heckscher–Ohlin model. Stolper and Samuelson (1941) stated: “The vast majority of
writers take it as axiomatic that a calculation of effects upon real income must take
into consideration the behaviour of prices of commodities entering the consumer’s
budget.” They saw their criterion which removed the index number problem as a great
advantage. But the criterion was quite inadequate for more general models. It was too
strict a test of real income change. Ironically, progress has been made in completing
the row and column properties of a general version of the theorem only by putting the
consumption effect back into the criterion.
References
Cassing, James, “On the Relationship between Commodity Price Changes and Factor Owners’
Real Positions,” Journal of Political Economy 89 (1981):593–5.
Chipman, John S., “Factor Price Equalization and the Stolper–Samuelson Theorem,” International Economic Review 10 (1969):399–406.
Deardorff, Alan V., “Overview of the Stolper–Samuelson Theorem,” in Alan V. Deardorff and
Robert M. Stern (eds.), The Stolper–Samuelson Theorem: A Golden Jubilee, Ann Arbor: University of Michigan Press (1994).
Deardorff, Alan V. and Robert M. Stern (eds.), The Stolper–Samuelson Theorem: A Golden
Jubilee, Ann Arbor: University of Michigan Press (1994).
Ethier, Wilfred J., “Some of the Theorems of International Trade with Many Goods and Factors,”
Journal of International Economics 4 (1974):199–206.
Helpman, Elhanan and Paul R. Krugman, Market Structure and Foreign Trade, Brighton: Wheatsheaf Books (1985).
Inada, K., “The Production Coefficient and the Stolper–Samuelson Condition,” Econometrica
39 (1971):219–40.
Jehle, G. A., Advanced Microeconomic Theory, Englewood Cliffs, NJ: Prentice-Hall (1991).
Jones, Ronald W., “The Structure of Simple General Equilibrium Models,” Journal of Political
Economy 73 (1965):557–72.
———, “A Three-Factor Model in Theory, Trade and History,” in Jagdish Bhagwati et al., Trade,
Balance of Payments and Growth, New York: McGraw-Hill (1971).
———, “Income Distribution and Effective Protection in a Multicommodity Trade Model,”
Journal of Economic Theory 11 (1975):1–15.
———, “Factor Scarcity, Factor Abundance and Attitudes towards Protection: the 3 ¥ 3 Model,”
Journal of International Economic Integration 7 (1992):1–19.
Jones, Ronald W. and T. Mitra, “Share Ribs and Income Distribution,” Review of International
Economics 3 (1995):36–52.
Jones, Ronald W. and Jose Scheinkman, “The Relevance of the Two-Sector Production Model
in Trade Theory,” Journal of Political Economy 85 (1977):909–35.
Kemp, Murray C. and Leon Wegge, “On the Relationship between Commodity Prices and Factor
Rewards,” International Economic Review 10 (1969):407–13.
Leamer, Edward, “Paths of Development in the Three-Factor n-Good General Equilibrium
Model,” Journal of Political Economy 95 (1987):961–99.
Lloyd, P. J. and A. G. Schweinberger, “Trade Expenditure Functions and the Gains from Trade,”
Journal of International Economics 24 (1988):275–97.
———, “Conflict-Generating Product Price Changes: The Imputed Output Approach,”
European Economic Review 41 (1997):1569–88.
Meade, James E., “Memorandum,” private correspondence reproduced in M. C. Kemp, Three
Topics in the Theory of International Trade: Distribution, Welfare and Uncertainty, Amsterdam:
North-Holland (1968).
© Blackwell Publishers Ltd 2000
GENERALIZING THE STOLPER–SAMUELSON THEOREM
613
Ruffin, Roy, “International Factor Movements,” in Ronald W. Jones and Peter B. Kenen (eds.),
Handbook of International Economics, Vol. 1, Amsterdam: North-Holland (1984).
Ruffin, Roy and Ronald Jones,“Protection and Real Wages: the Neoclassical Ambiguity,” Journal
of Economic Theory 14 (1977):337–48.
Shimomura, K., “A Geometric Approach to the Stolper–Samuelson Theorem,” International
Economic Review 38 (1997):647–56.
Stolper, Wolfgang F. and Paul A. Samuelson, “Protection and Real Wages,” Review of Economic
Studies 9 (1941):58–73.
Uekawa, Y., “Generalization of the Stolper–Samuelson Theorem,” Econometrica 39 (1971):197–
217.
Notes
1. Others have been proposed but they are less interesting. For example, Inada (1971) proposed
a sign matrix in which the diagonal elements are negative and all the off-diagonal elements are
positive, but this is a mathematical curiosity.
2. The model can be extended to some situations with economies of scale which are external to
the firm and joint production (see Lloyd and Schweinberger, 1997). It can also be extended to
a labor–leisure choice, using the concept of full income, and to ad valorem commodity taxes
which maintain a fixed wedge between producer and consumer prices, and to ad valorem factor
taxes.
3. Alternatively, from equation (15), sign [V ] = sign [E ], where E is the matrix whose elements
are the household excess supply (e hi ). The rows of this matrix must have a positive and a negative element since Sipie hi = 0.
4. I am indebted to a referee for pointing out this case.
5. This interpretation does not hold for the case of m < n as the output functions are not differentiable in this case.
6. In the n ¥ n case, the full-employment conditions are A(w)y = v t. Hence, the outputs are given
by y = [A(w)]-1vt. If households have the same technology the household outputs are given by
ỹ h = [A(w)]-1vht.
7. This generalizes a result which has been known to apply to an exchange economy (e.g., Jehle,
1991, p. 350) to an economy with production. It also extends to the household unit a result which
is known to apply to a small price-taking nation which trades with other nations.
8. The easiest way to obtain the Rybczynski terms is to use the Jones hat calculus in the manner
suggested by Jones (1975).
© Blackwell Publishers Ltd 2000